I have a linear program where the variables are n vectors. Now I'd like to impose an extra constraint that k (k<=n) of the n vectors are linearly independent, or the matrix with the n vectors as rows has rank k. It seems to me that this constraint destroys the linearity. Is there any existing theory dealing with this kind of problem?

1 Answer
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Let $k = 1$. Then all your vectors are multiples of the same vector. Number them $v_0,\ldots,v_n$. Add the constraints $v_{00} = 1$ and $v_{i0} = v_{0i}$. The rank constraint implies that $v_{ij}/v_{i0} = v_{0j}/v_{00}$ and so $v_{ij} = v_{i0} v_{0j} = v_{i0} v_{j0}$. Put $x_i = v_{i0}$, so that . We thus have a quadratically constrained quadratic program, which is NP-hard to optimize. The reason is that we can simulate zero/one integer programming by adding the constraint $x_i^2 = x_i$.